Linear combination of terms of stationary process is stationary

Question

I'm just learning about stationary stochastic processes and I'm a little confused about one example. If $X = (X_t)_{t \in I}$ is a stochastic process with values in a Borel space $(E, \mathcal E)$, its distribution $\mathcal L[X]$ is a measure on the space $E^I$ of functions $f : I \to E$ with $\sigma$-algebra generated by sets of the form $\prod_{t \in I} A_t$ with $A_t \in \mathcal E$, and $A_t = E$ for all but countably many $t$. In particular, if $A = \prod_{t \in I}A_t$, we have $$ \mathcal L[X] \left( A \right) = \mathbb P[X \in A] = \mathbb P\left[ \mathop\bigcap_{t \in I} \left\{X_t \in A_t\right\} \right] $$ Suppose $X = (X_n)_{n \in \mathbb Z}$ is a real-valued stationary process, meaning $\mathcal L[(X_{s+t})_{t \in I}] = \mathcal L[(X_t)_{t \in I}]$ for every $s \in I$, and let $k \in \mathbb N$ and $c_0, \ldots, c_k \in \mathbb R$. Define the stochastic process $Y = (Y_n)_{n \in \mathbb Z}$ by $$ Y_n = \sum_{i=0}^k c_i X_{n-i} $$ How can I show that $Y$ is stationary? I tried directly applying the above formula for $\mathcal L[(Y_{n+m})_{n \in \mathbb Z}]$ but it hasn't gotten me very far. Any tips?

Answer 1

Here are two lemmas which should serve both as useful tools and as suggestive hints.

Lemma 1: If $X$ is an $E$-valued stationary process, then so is the $E^{k+1}$ valued process $\vec X_n:= (X_n,X_{n-1},...,X_{n-k})$.

Proof Outline: Consider measurable sets $A_1,...,A_r$ in $E^k$ of the form $A_i=A_{1,i} \times \cdots \times A_{k,i}$, where $A_{j,i} \subset E$. Let $\theta: (E^{k+1})^{\Bbb N} \to (E^{k+1})^{\Bbb N}$ be the shift operator $(\vec x_n)_n \mapsto (\vec x_{n+1})_n$. Let $A = \prod_{i \in \Bbb N} A_i$, where $A_i:=E$ for $i>r$. Show that $P^{\vec X}(\theta^{-1}(A)) = P^{\vec X}(A)$, where $P^{\vec X}$ denotes the law of $\vec X$ on $(E^{k+1})^{\Bbb N}$. This is where you'll need stationarity. Conclude the same for general $A$ using $\pi$-$\lambda$ or monotone class theorems. $\Box$

Lemma 2: If $X$ is an $E$-valued stationary process and if $f:E \to E'$ is a measurable map (where $E'$ is some other measurable space), then $f(X)$ is a stationary $E'$-valued process.

Proof Outline: Again one considers measurable sets in $(E')^{\Bbb N}$ which are of the form $A_1 \times \cdots \times A_n \times E \times E \times \cdots$. On such sets the laws of $f(X)$ and of $\theta f(X)$ will agree. Then one concludes using $\pi$-$\lambda$ again. $\Box$